Finite Group Databasebeta

← #29828831

Subgroups — #29828831

Order 15616.

1868 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D4×Dic61C211yesmaximal, characteristic
2C2×#16839965C212yesmaximal, complemented
3C2×#16839913C211yesmaximal, characteristic
4#21454167C211yesmaximal, characteristic, complemented
5#21446155C212yesmaximal, complemented
6#21446151C212yesmaximal, complemented
7#21446127C211yesmaximal, characteristic, complemented
8#21446124C211yesmaximal, characteristic, complemented
9#21445817C212yesmaximal, complemented
10#21445797C211yesmaximal, characteristic, complemented
11C2×C61×#450C211yesmaximal, characteristic
12C24×Dic61—21no—
13C22×#11758200C2212yes—
14C2×#14519775C2211yescharacteristic, complemented
15C2×#14519756C2211yescharacteristic
16C2×D4×Dic61—28no—
17C2×D4×Dic61—21no—
18C2×D4×Dic61—21no—
19C2×D4×Dic61C2212yes—
20C2×D4×Dic61C2212yes—
21C22×C4⋊Dic61C2211yescharacteristic
22C22×C22⋊Dic61—21no—
23C22×C4×Dic61C2211yescharacteristic
24#16839965—28no—
25#16839913—24no—
26C2×#14519706C2212yescomplemented
27C2×#14519702C2212yescomplemented
28C2×#14519701C2212yescomplemented
29C2×#14519698C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31#16839669C2211yescharacteristic, complemented
32#16839667C2211yescharacteristic, complemented
33#16839657—21no—
34#16839657—21no—
35#16839631C2211yescharacteristic, complemented
36#16839620—21no—
37#16839620—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839516C2212yescomplemented
41C23×D4×C61C2211yescharacteristic
42C22×SD16×C61C2212yes—
43C2×C22⋊Q8×C61C2211yescharacteristic
44C61×#450—24no—
45C2×#102×C61C2212yes—
46C2×C22⋊C8×C61C2211yescharacteristic
47C22×Dic122C2311yescharacteristic
48C23×Dic61—28no—
49C23×Dic61—21no—
50C23×Dic61—22no—
51C23×Dic61—22no—
52C23×Dic61—44no—
53C23×Dic61—44no—
54C23×Dic61D411yescharacteristic
55C23×Dic61D411yescharacteristic
56C2×#11758200—24no—
57C2×#11758200—24no—
58C2×#11758200—24no—
59C22×#92418C2312yes—
60#14519775—24no—
61#14519756—22no—
62#14519756—22no—
63D4×Dic61—28no—
64D4×Dic61—28no—
65D4×Dic61—24no—
66D4×Dic61—24no—
67D4×Dic61—24no—
68D4×Dic61D418yes—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—24no—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2312yes—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×C22⋊Dic61—28no—
79C2×C22⋊Dic61—22no—
80C2×C22⋊Dic61—22no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—24no—
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D411yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519706—24no—
88#14519702—24no—
89#14519701—24no—
90#14519698—24no—
91C22⋊#92418—22no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96C25×C61—21no—
97C22×Q8×C61C2311yescharacteristic
98C22×D4×C61—24no—
99C22×D4×C61—24no—
100C22×D4×C61—21no—
101C22×D4×C61—22no—
102C22×D4×C61D414yes—
103C22×D4×C61C2312yes—
104C23×C4×C61C2311yescharacteristic
105C2×SD16×C61—24no—
106C2×SD16×C61—24no—
107C2×SD16×C61—24no—
108C22×C8×C61C2312yes—
109C22⋊Q8×C61—22no—
110C22⋊Q8×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C22⋊C4×C61—21no—
114#102×C61—24no—
115C22⋊C8×C61—22no—
116C24×C61—24no—
117C24×C61—24no—
118C24×C61—24no—
119C24×C61—22no—
120C24×C61—42no—
121C24×C61—42no—
122C24×C61—42no—
123C24×C61—42no—
124C24×C61C2×D411yescharacteristic
125C2×Q8×C61—21no—
126C2×Q8×C61—22no—
127C2×Q8×C61—21no—
128C2×Q8×C61—22no—
129C2×D4×C61—24no—
130C2×D4×C61—216no—
131C2×D4×C61—24no—
132C2×D4×C61—216no—
133C2×D4×C61—24no—
134C2×D4×C61—22no—
135C2×D4×C61—22no—
136C2×D4×C61—22no—
137C2×D4×C61C2×D414yes—
138C2×D4×C61C2×D414yes—
139C2×D4×C61#4114yes—
140C22×C4×C61—22no—
141C22×C4×C61—22no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D412yes—
146C22×C4×C61#4112yes—
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—22no—
149C22×Dic61—22no—
150C22×Dic61—24no—
151C22×Dic61—28no—
152C22×Dic61—22no—
153C22×Dic61—22no—
154C22×Dic61—28no—
155C22×Dic61—22no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—41no—
159C22×Dic61—44no—
160C22×Dic61—432no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—44no—
164C22×Dic61—44no—
165C22×Dic61—44no—
166C22×Dic61SD1614yescomplemented
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C2×Dic122—22no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—22no—
175SD16×C61—48no—
176C2×C8×C61—22no—
177C2×C8×C61—22no—
178C2×C8×C61—22no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—42no—
182C22⋊C4×C61—42no—
183C22⋊Dic61—28no—
184C22⋊Dic61—28no—
185C22⋊Dic61—42no—
186#11758200—48no—
187C4⋊Dic61—24no—
188C4⋊Dic61—22no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61C2×D414yes—
192#11758196—42no—
193#11758196—42no—
194#11758196—42no—
195#11758196—42no—
196#11758196—42no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61C2×D414yes—
201C2×#92418—22no—
202C2×#92418—22no—
203C2×#92418—22no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61—216no—
209C23×C61—24no—
210C23×C61—24no—
211C23×C61—24no—
212C23×C61—48no—
213C23×C61—48no—
214C23×C61—48no—
215C23×C61—48no—
216C23×C61—48no—
217C23×C61—42no—
218C23×C61—42no—
219C23×C61—42no—
220C23×C61—42no—
221C23×C61—48no—
222C23×C61#12112yes—
223C23×C61C2×SD1612yes—
224C23×C61#13712yes—
225C23×C61C22×D411yescharacteristic
226Q8×C61—42no—
227Q8×C61—42no—
228D4×C61—216no—
229D4×C61—24no—
230D4×C61—24no—
231D4×C61—24no—
232D4×C61#12118yes—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—24no—
236C2×C4×C61—24no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61#12112yes—
244C2×C4×C61C22×D411yescharacteristic
245C2×C4×C61C22×D412yes—
246C2×C4×C61C2×#4112yes—
247C2×C4×C61C2×#4111yescharacteristic
248C2×Dic61—21no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—28no—
253C2×Dic61—21no—
254C2×Dic61—24no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—44no—
258C2×Dic61—416no—
259C2×Dic61—48no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—416no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61#12012yes—
270C2×Dic61#12012yes—
271C2×Dic61C2×SD1612yes—
272C2×Dic61C2×SD1612yes—
273C2×Dic61AGL(1,ℤ/8)12yes—
274Dic122—42no—
275Dic122—42no—
276C8×C61—42no—
277#92418—42no—
278#32905—611nomaximal, Sylow
279C22×C61—22no—
280C22×C61—22no—
281C22×C61—22no—
282C22×C61—22no—
283C22×C61—22no—
284C22×C61—22no—
285C22×C61—28no—
286C22×C61—432no—
287C22×C61—44no—
288C22×C61—44no—
289C22×C61—44no—
290C22×C61—44no—
291C22×C61—44no—
292C22×C61—44no—
293C22×C61#46114yescomplemented
294C22×C61C2×#12011yescharacteristic
295C22×C61C2×#12111yescharacteristic
296C22×C61#52511yescharacteristic
297C22×C61C22×SD1611yescharacteristic
298C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
299C22×C61C2×#13711yescharacteristic
300C22×C61#57511yescharacteristic
301C4×C61—22no—
302C4×C61—81no—
303C4×C61—81no—
304C4×C61C2×#12112yes—
305C4×C61#52512yes—
306Dic61—22no—
307Dic61—22no—
308Dic61—44no—
309Dic61—41no—
310Dic61—81no—
311Dic61—81no—
312Dic61#45014yescomplemented
313C22×C4×D4—611no—
314C2×#461—612no—
315C2×#450—611no—
316C2×#450D6111yescharacteristic
317#2857—611no—
318#1877—612no—
319#1877—612no—
320#1872—611no—
321#1872—611no—
322#1802—612no—
323#1797—611no—
324C2×C61—24no—
325C2×C61—48no—
326C2×C61#284711yescharacteristic
327C2×C61C2×#45011yescharacteristic
328C2×C61#300711yescharacteristic
329C2×C61C2×#46111yescharacteristic
330C2×C61#303611yescharacteristic
331C2×C61#310111yescharacteristic
332C2×C61#311011yescharacteristic
333C23×D4C2×D6111yescharacteristic
334C24×C4—1221no—
335C22×SD16—612no—
336C22×SD16C2×D6112yes—
337C2×C4⋊Q8—611no—
338C2×C22⋊Q8—611no—
339C2×C22⋊Q8C2×D6111yescharacteristic
340C2×C4×D4—612no—
341C2×C4×D4—612no—
342C2×C4×D4—1221no—
343C2×C4×D4—1221no—
344C2×C4×D4—1228no—
345C22×C4⋊C4—611no—
346C22×C22⋊C4—1221no—
347C22×C42—611no—
348#461—1228no—
349#450—24no—
350#450—1224no—
351C2×C4⋊C8—612no—
352C2×#103—612no—
353C2×#103—612no—
354C2×#102—612no—
355C2×#102C2×D6112yes—
356C2×C22⋊C8—611no—
357C2×C22⋊C8C2×D6111yescharacteristic
358#391—611no—
359#391—611no—
360#389—1221no—
361#389—1221no—
362#384—611no—
363#380—1221no—
364#380—1221no—
365#340—612no—
366#340—612no—
367#336—612no—
368C61#3290511yescharacteristic, complemented, Sylow
369C25—21no—
370C22×Q8—611no—
371C22×Q8C22×D6111yescharacteristic
372C22×D4—24no—
373C22×D4—24no—
374C22×D4—21no—
375C22×D4—22no—
376C22×D4C22⋊D6114yes—
377C22×D4C22×D6112yes—
378C23×C4—611no—
379C23×C4—611no—
380C23×C4—1222no—
381C23×C4—1222no—
382C23×C4—1228no—
383C23×C4—1221no—
384C23×C4—2444no—
385C23×C4—2444no—
386C23×C4C22×D6111yescharacteristic
387C2×SD16—24no—
388C2×SD16—24no—
389C2×SD16—24no—
390C2×SD16—1224no—
391C2×SD16—1224no—
392C2×SD16—1224no—
393C22×C8—612no—
394C22×C8C22×D6112yes—
395C4⋊Q8—1224no—
396C22⋊Q8—22no—
397C22⋊Q8—22no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400C4×D4—618no—
401C4×D4—1224no—
402C4×D4—1224no—
403C4×D4—1228no—
404C4×D4—1228no—
405C4×D4—1224no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—1224no—
410C2×C4⋊C4—1221no—
411C2×C4⋊C4—1221no—
412C2×C4⋊C4—1221no—
413C2×C4⋊C4—1221no—
414C2×C4⋊C4—1221no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4C22×D6111yescharacteristic
417C2×C22⋊C4—21no—
418C2×C22⋊C4—1222no—
419C2×C22⋊C4—1222no—
420C2×C22⋊C4—1228no—
421C2×C22⋊C4—1221no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—1224no—
427C4⋊C8—1224no—
428#103—1224no—
429#103—1224no—
430#102—24no—
431#102—1224no—
432C22⋊C8—22no—
433C22⋊C8—1222no—
434#95—1221no—
435#95—1221no—
436#95—1221no—
437#95—1221no—
438C24—24no—
439C24—24no—
440C24—22no—
441C24—24no—
442C24—42no—
443C24—42no—
444C24—42no—
445C24—42no—
446C24D4×D6111yescharacteristic
447C2×Q8—22no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×Q8—1222no—
452C2×Q8—1221no—
453C2×Q8—1221no—
454C2×Q8—1222no—
455C2×D4—24no—
456C2×D4—24no—
457C2×D4—216no—
458C2×D4—216no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—24no—
463C2×D4D4×D6114yes—
464C2×D4#1175822414yes—
465C2×D4C2×C22⋊D6114yes—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—614no—
474C22×C4—611no—
475C22×C4—1222no—
476C22×C4—1222no—
477C22×C4—1222no—
478C22×C4—1228no—
479C22×C4—1224no—
480C22×C4—1228no—
481C22×C4—1222no—
482C22×C4—1222no—
483C22×C4—1221no—
484C22×C4—1221no—
485C22×C4—2444no—
486C22×C4—2444no—
487C22×C4—2444no—
488C22×C4—2441no—
489C22×C4—24432no—
490C22×C4—2444no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4D4×D6111yescharacteristic
494C22×C4#1175822412yes—
495C22×C4C2×C22⋊D6112yes—
496C22×C4C23×D6111yescharacteristic
497SD16—48no—
498SD16—2448no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—1222no—
503C2×C8—1222no—
504C2×C8—1222no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—42no—
508C4⋊C4—614no—
509C4⋊C4—1224no—
510C4⋊C4—1222no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—2442no—
514C4⋊C4—2442no—
515C4⋊C4—2442no—
516C4⋊C4—2442no—
517C4⋊C4—2442no—
518C22⋊C4—42no—
519C22⋊C4—1228no—
520C22⋊C4—1228no—
521C22⋊C4—2442no—
522C42—614no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—216no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—48no—
535C23—48no—
536C23—48no—
537C23—48no—
538C23—48no—
539C23—48no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23#1451974912yes—
545C23SD16×D6112yes—
546C23#1451980612yes—
547C23C2×D4×D6111yescharacteristic
548Q8—42no—
549Q8—42no—
550Q8—2442no—
551Q8—2442no—
552D4—216no—
553D4—24no—
554D4—24no—
555D4—24no—
556D4#1451974918yes—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—612no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—1228no—
573C2×C4—1221no—
574C2×C4—1221no—
575C2×C4—1221no—
576C2×C4—1221no—
577C2×C4—1221no—
578C2×C4—1224no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—24416no—
582C2×C4—2444no—
583C2×C4—24416no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4—2448no—
591C2×C4—2442no—
592C2×C4—2442no—
593C2×C4#1451974912yes—
594C2×C4C2×D4×D6112yes—
595C2×C4C22×C22⋊D6111yescharacteristic
596C2×C4C2×#1175822412yes—
597C2×C4C2×#1175822411yescharacteristic
598C8—42no—
599C8—2442no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—28no—
607C22—432no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22#1683996514yescomplemented
615C22#120×D6111yescharacteristic
616C22C2×#1451974911yescharacteristic
617C22#1684026811yescharacteristic
618C22C2×SD16×D6111yescharacteristic
619C22AGL(1,ℤ/8)×D6111yescharacteristic
620C22C2×#1451980611yescharacteristic
621C22#1684055811yescharacteristic
622C4—22no—
623C4—81no—
624C4—81no—
625C4—614no—
626C4—1222no—
627C4—1222no—
628C4—4881no—
629C4—4881no—
630C4—2444no—
631C4—2441no—
632C4C2×#1451974912yes—
633C4#1684026812yes—
634C2—24no—
635C2—48no—
636C2#2145409911yescharacteristic
637C2#450×D6111yescharacteristic
638C2#2145507411yescharacteristic
639C2C2×#1683996511yescharacteristic
640C2#2145528011yescharacteristic
641C2#2145582211yescharacteristic
642C2#2145591311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#450C2
socleC23×C61C22×D4
prime radicalC23×D4×C61C22
hypercenterC2×#450D61
hypocenterC61#32905
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#450

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16839965C2semidirect
4. #21454167C2semidirect
5. #21446155C2semidirect
6. #21446151C2semidirect
7. #21446127C2semidirect
8. #21446124C2semidirect
9. #21445817C2semidirect
10. #21445797C2semidirect
14. C2×#14519775C22semidirect
26. C2×#14519706C22semidirect
27. C2×#14519702C22semidirect
28. C2×#14519701C22semidirect
31. #16839669C22semidirect
32. #16839667C22semidirect
35. #16839631C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839516C22semidirect
166. C22×Dic61SD16semidirect
293. C22×C61#461semidirect
312. Dic61#450semidirect
368. C61#32905semidirect
614. C22#16839965semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×D4×Dic612
C2×#168399652
C2×#168399132
#214541672
#214461552
#214461512
#214461272
#214461242
#214458172
#214457972
C2×C61×#4502
#3290561